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Elliptic operators in subspaces and the eta invariant. (English) Zbl 1021.58017
Let \(A\) be a self-adjoint elliptic operator on a closed manifold \(M\) of dimension \(m\). One defines \(\eta(s,A):=\sum_\nu sign(\lambda_\nu)|\lambda_\nu|^{-s}\) as a measure of the spectral asymmetry of \(A\). This has an analytic continuation to \(C\); \(s=0\) is a regular value and one sets \(\eta(A)=\eta(0,A)/2\). If \(A_t\) is a smooth \(1\) parameter family of partial differential operators of order \(r\) and \(r+m\) is odd, then the mod \(Z\) reduction of \(\eta(A_t)\) is independent of the parameter \(t\).
The authors derive a formula for this fractional part of the eta invariant in topological terms using the index theorem for elliptic operators on subspaces. The authors also study the \(K\)-theory of \(M\) with coefficients in \(Z_n\).
The authors have used the results of this paper in subsequent work to construct some new second order even order operators in odd dimensions with non-trivial eta invariant.

58J28 Eta-invariants, Chern-Simons invariants
58J20 Index theory and related fixed-point theorems on manifolds
58J22 Exotic index theories on manifolds
19K56 Index theory
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